The Northside Bank is working to develop an efficient work schedule for full$-$time and part$-$time tellers. The schedule must provide for efficient operation of the bank, including adequate customer service, employee breaks, and so on$.$ On Fridays, the bank is open from $9$:$00$ a$.$m$.$ to $7$:$00$ p$.$m$.$ The number of tellers necessary to provide adequate customer service during each hour of operation is summarized as follows.

$\backslash$table$[[$Time$,\backslash$table$[[$No$.$ of$],[$Tellers$]]],[9$:$00$ a$.$m$.-10$:$00$ a$.$m$.,6],[10$:$00$ a$.$m$.-11$:$00$ a$.$m$.,4],[11$:$00$ a$.$m$.-$ Noon,$8],[$Noon $-1$:$00$ p$.$m$.,10],[1$:$00$ p$.$m$.-2$:$00$ p$.$m$.,9],[2$:$00$ p$.$m$.-3$:$00$ p$.$m$.,6],[3$:$00$ p$.$m$.-4$:$00$ p$.$m$.,4],[4$:$00$ p$.$m$.-5$:$00$ p$.$m$.,7],[5$:$00$ p$.$m$.-6$:$00$ p$.$m$.,6],[6$:$00$ p$.$m$.-7$:$00$ p$.$m$.,6]]$

Each full$-$time employee starts on the hour and works a $4-$hour shift, followed by a $1-$hour break and then a $3-$hour shift. Parttime employees work one $4-$hour shift beginning on the hour. Considering salary and fringe benefits, full$-$time employees cost the bank $$14$ per hour $($ $$98$ a day$),$ and part$-$time employees cost the bank $$7$ per hour $($ $$28$ per day$).$

$($a$)$ Formulate an integer programming model that can be used to develop a schedule that will satisfy customer service needs at a minimum employee cost $($in $ per day$).($Let $x_i=$ number of full$-$time employees coming on duty at the beginning of hour i and $y_i=$ number of part$-$time employees coming on duty at the beginning of hour i where $i=9,10,11,12,1,2,3,4,5,6.$ Assume that each employee must be able to finish their complete shift before the bank closes.$)$

Min

$9$:$00a.m.-10$:$00a.m.$

$10$:$00a.m.-11$:$00a.m.$

$11$:$00a.m.-$ Noon $$

Noon $-1$:$00p.m.$

$1$:$00p.m.-2$:$00p.m.$

$2$:$00p.m.-3$:$00p.m.$

$3$:$00p.m.-4$:$00p.m.$

$4$:$00p.m.-5$:$00p.m.$

$5$:$00p.m.-6$:$00p.m.$

$6$:$00p.m.-7$:$00p.m.$

$10$:$00$ a$.$m$.-11$:$00$ a$.$m$..$m$.($b$)$ Solve the LP relaxation of your model in part $($a$).$Solving the LP relaxation tells us to use a total of $$ full$-$time employees and $$ part$-$time employees at a cost of $ per day.$($c$)$ Solve your model in part $($a$)$ for the optimal schedule of tellers.Solving for the optimal schedule tells us to use a total offull$-$time employees and$|$ part$-$time employees at a cost of $ per day.Comment on the solution.This solution may not be ideal for the Northside Bank since the cost per der day is over $$1,000.$This solution may not be ideal for the Northside Bank since it only makes use of full$-$time employees.This solution seems very cost effective for the Northside Bank since the cost per day is below $$500.$This solution seems reasonable since it requires the Northside Bank to use a near even mix of part$-$time and fulltime employees.This solution may not be ideal for the Northside Bank since it only makes use of part$-$time employees.

$($d$)$ After reviewing the solution to part $($c$),$ the bank manager realized that some additional requirements must be specified. Specifically, she wants to ensure that one full$-$time employee is on duty at all times and that there is a staff of at least five full$-$time employees on Fridays. Revise your model to incorporate these additional requirements.

In addition to the constraints from part $($a$),$ what additional constraints should be added to the integer programming model?

Full$-$Time during $9$:$00$ a$.$m$.-10$:$00$ a$.$m$.$

Full$-$Time during $10$:$00$ a$.$m$.-11$:$00$ a$.$m$.$

Full$-$Time during $11$:$00$ a$.$m$.-$ Noon

Full$-$Time during Noon $-1$:$00$ p$.$m$.$

Full$-$Time during $1$:$00$ p$.$m$.-2$:$00$ p$.$m$.$

Full$-$Time during $2$:$00$ p$.$m$.-3$:$00$ p$.$m$.$

Full$-$Time during $3$:$00$ p$.$m$.-4$:$00$ p$.$m$.$

Full$-$Time during $4$:$00$ p$.$m$.-5$:$00$ p$.$m$.$

Full$-$Time during $5$:$00$ p$.$m$.-6$:$00$ p$.$m$.$

Full$-$Time during $6$:$00$ p$.$m$.-7$:$00$ p$.$m$.$

Total Full$-$Time Employees

Solve for the optimal solution.

Solving for the optimal schedule tells us to use a total of full$-$time employees and part$-$time employees at a cost of $ per day.